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Theorem imass1 6100
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
Assertion
Ref Expression
imass1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem imass1
StepHypRef Expression
1 ssres 6008 . . 3 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 rnss 5938 . . 3 ((𝐴𝐶) ⊆ (𝐵𝐶) → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
31, 2syl 17 . 2 (𝐴𝐵 → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
4 df-ima 5689 . 2 (𝐴𝐶) = ran (𝐴𝐶)
5 df-ima 5689 . 2 (𝐵𝐶) = ran (𝐵𝐶)
63, 4, 53sstr4g 4027 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3948  ran crn 5677  cres 5678  cima 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  predrelss  6338  vdwnnlem1  16927  dprdres  19897  imasnopn  23193  imasncld  23194  imasncls  23195  utoptop  23738  restutop  23741  ustuqtop3  23747  utopreg  23756  metustbl  24074  imadifxp  31827  esum2d  33086  eulerpartlemmf  33369  bj-imdirco  36066  brtrclfv2  42468  frege97d  42493  frege109d  42498  frege131d  42505  hess  42521  resimass  43933  setrecsss  47736
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